Strong relaxation limit and uniform time asymptotics of the Jin-Xin model in the $L^p$ framework

Abstract

We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in ${\mathbb{R}}^d$ for $d\ge 1$. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter $\epsilon >0$, for initial data in hybrid Besov spaces based on $L^p$-norms. This uniformity enables us to derive $\mathcal{O}(\epsilon )$ bounds on the difference between solutions of the viscous conservation law and its associated Jin-Xin approximation, thus justifying the strong convergence of the relaxation process. Furthermore, under an additional condition on the initial data, for instance, that the low frequencies belong to $L^{p/2}({\mathbb{R}}^d)$, we show that the $L^p({\mathbb{R}}^d)$-norm of the solution to the Jin-Xin model decays at the optimal rate $(1+t{)}^{-d/2p}$, and the $L^p({\mathbb{R}}^d)$-norm of its difference with the solution of the associated viscous conservation law decays at the enhanced rate $\epsilon (1+t{)}^{-d/2p-1/2}$.

38 pages, to appear in SCIENCE CHINA Mathematics